AI 中文总结
该研究推导了负二项分布与超几何分布在众数附近的完全局部渐近展开，补充了二项分布的相关结果，明确了系数形式与众数规则，为离散分布的局部渐近分析提供了完善。
AI 中文摘要
我们推导了负二项分布与超几何分布在众数附近的完全局部渐近展开，补充了姊妹论文中得到的二项分布展开。系数以封闭的伯努利多项式形式给出，保留了相对于均值的精确晶格位移。对于每种分布，熵归一化将伯努利多项式替换为倒数伯努利多项式，使自然反射对称性显现。一阶系数还恢复了精确众数规则：在卡茨情形中，顶点恰好给出经典阈值；而在超几何情形中，其显式偏差始终过小，不会改变所选整数众数。
英文摘要
We derive complete local asymptotic expansions near the mode for the negative binomial and hypergeometric laws, complementing the binomial expansion obtained in the companion papers. The coefficients are given in closed Bernoulli-polynomial form and retain the exact lattice displacement from the mean. For each law an entropy normalisation replaces the Bernoulli polynomials by reciprocal Bernoulli polynomials and makes the natural reflection symmetry visible. The first-order coefficient also recovers the exact mode rule: in the Katz cases the vertex gives the classical threshold exactly, while in the hypergeometric case its explicit discrepancy is always too small to change the selected integer mode.
Comments15 pages
